Waist on a torus

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Waist on a torus

November 22, 2011 mymbl Leave a comment Go to comments

Last year I worked on a interesting problem of the possibility of partitioning a convex set in a plane in to ${n}$ internally disjoint convex pieces of equal area and perimeter. The problem has since been proved when ${n}$ is a prime power, by Aronov and Hubard and Roman Karasev for all dimensions. Here in this post I discuss a crucial lemma i worked while attempting to prove the n is a “power of two” case. In the simplest case, the lemma goes like this: given a continuous function ${f:(S^{1})^{2} \rightarrow \mathrm{I\!R}}$ which has the ${y}$ -antipodal property: ${f(x,-y) = -f(x,y)}$ , there is a pair of separators of a torus along its “waist” that map to zero under ${f}$ . More specifically, there is a subset ${\Lambda}$ of the zero set of ${f}$ which is such that the projection map ${p_1(x,y) = x}$ restricted to ${\Lambda}$ is not null homotopic. For larger powers of ${2}$ , the notation gets a little unwieldy but the proof scales up well. Lets tackle notation first:

Label the coordinates of ${(S^1)^{2^n}}$ in a binary system, with words using the letters ${0}$ , ${1}$ and ${2}$ . The words that can be formed using these letters can be treated as vertices in a binary tree, with the word ${0}$ as the root node. The root node has a unique child vertex as the word ${1}$ . Other vertices consist of words ${w}$ in letters ${\{1,2\}}$ ( of length ${<n}$ ) each having child nodes ${w1}$ and ${w2}$ . For example, the binary tree for ${n=3}$ would have vertices as below:

If we impose the dictionary ordering on such words, ${0 < 1 < 2 < 11 < 12 < 21 < 22 < 111 < 222 \ldots, 12\ldots2}$ , then each word of length ${l \le n}$ uniquely identifies a coordinate in ${(S^1)^{2^n}}$ . For example, a point ${x}$ in ${S^{8}}$ can be written as ${(x_0,x_1,x_{11},x_{12},x_{111},x_{112},x_{121},x_{122})}$ .

Let ${W}$ be the set of words representing indices to coordinates of points in ${(S^1)^{2^n}}$ that have letters in 0,1 and 2, with 0 appearing only as the starting letter and are of length not more than ${n}$ . For each word ${w \in W}$ of length ${k\le n}$ we define ${p_{w}:(S^1)^{2^n} \rightarrow S}$ to be the projection map: ${p_{w}(x_0,x_1,\ldots,x_{w},\ldots) = x_{w}}$ . Given ${w\in W}$ , we define a map ${\omega_{w}: (S^1)^{2^n} \rightarrow (S^1)^{2^n}}$ as follows:

$\displaystyle p_{w1w'}(\omega_{w}(x)) = x_{w2w'}, \text{ for all words} \quad w1w' \in W$

$\displaystyle p_{w2w'}(\omega_{w}(x)) = x_{w1w'}, \text{ for all words} \quad w2w' \in W$

$\displaystyle p_w(\omega_w(x)) = -x_w$

and for every other word ${v \in W}$ , ${p_v( \omega_w(x)) = x_v}$ .

For instance, for ${x \in S^{8}}$ and ${w = 1}$ , we have,

$\displaystyle \omega_{1}(x) = \omega_{1}(x_0,x_1,x_{11},x_{12},x_{111},x_{112},x_{121},x_{122}) = (x_0,-x_1,x_{12},x_{11},x_{121},x_{122},x_{111},x_{112})$

Thus, ${\omega_{w}}$ changes the sign of the coordinate with index ${w}$ and swaps coordinates with indexes ${w1v}$ with ${w2v}$ , leaving all the other coordinates fixed. If ${w_k}$ represents a subword of ${w}$ consisting of first ${k}$ letters from ${w}$ , then using the definition of ${\omega_k}$ , we note in particular, that ${p_{w_k}( \omega_w(x)) = x_{w_k}}$ . Finally let ${\text{len}: W \rightarrow \mathrm{I\!N}}$ to be the map that gives the length of a word ${w}$ in ${W}$ .

Definition 1 ( ${w}$ -antipodal map) For ${w \in W}$ , we say that a map ${P:(S^1)^{2^n} \rightarrow \mathrm{I\!R}^{2^n-1}}$ is ${w}$ -antipodal, if,
$\displaystyle {\tilde p_w}(P(\omega_{w}(x))) = -{\tilde p_w}(P(x))$

.

Theorem 2 Let ${{f}: (S^1)^{2^n} \rightarrow \mathrm{I\!R}^{2^n-1}}$ be a continuous map with the ${w}$ -antipodal property for each word ${w \in W}$ . Suppose there exists ${\mathbf{c} \in (S^1)^{2^n}}$ such that ${f(\mathbf{c}) = \mathbf{0}}$ . Then the zero set of ${f}$ contains a connected subset ${\Lambda}$ which can be approximated arbitrarily closely in the Hausdorff distance by a smooth loop ${L}$ , for which the projection map: ${p_0 : L \rightarrow S^1}$ , ${p_0(y_0,y_1,\ldots) = y_0}$ is not null-homotopic.

The second lemma is a Borsuk Ulam type theorem which has a cleaner and simpler proof:

Lemma 2 Let ${P}$ be a real valued function defined on the connected closed subset ${\Lambda}$ on the torus ${(S^1)^{k}}$ such that projection ${p_1:S^1 \times (S^1)^{k-1} \rightarrow S^1}$ defined by ${p_1(x,\mathbf{y}) = x}$ is not null-homotopic. Then there exists two points ${(x,\mathbf{y_1})}$ and ${(-x,\mathbf{y_2})}$ in ${\Lambda}$ such that ${P(x,\mathbf{y_1}) = P(-x,\mathbf{y_2})}$ .